How to Solve Quadratic Equations

Let's look at how to solve quadratic equations. In addition to the properties of equations, knowledge of square roots and factorization is necessary because it's a quadratic expression.

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Approach to Solving

While we could find the solution by drawing and reading a graph as mentioned above, it's not always possible to draw a graph, and the intersections with the x-axis aren't always integers, so it's not practical to draw a graph every time to find the solution. Therefore, like with linear equations, we consider methods to find the solution by transforming the equation.

The above figure extracts the units that use letters and expressions to reveal unknowns within the scope of middle school mathematics. In this range, quadratic equations are the final boss. You can't solve quadratic equations without mastering the previous units.

Quadratic equations have a degree of 2 for $x$ , and to solve this equation, we need to handle this quadratic state skillfully. Among the units we've learned so far, squares appeared in the expansion and factorization of polynomials and square roots in the 9th grade. Also, regarding equations, we learned about linear equations in the 7th grade. Let's try solving quadratic equations using this knowledge.

Using Factorization

Factorization was about decomposing a number or polynomial into factors and expressing it as a product. Specifically, we learned to factorize quadratic polynomials into the form $(x+a)(x+b)$ . For example, $x^2+3x+2$ factorizes to $(x+1)(x+2)$ .

Can we apply this to quadratic equations? What if we could transform the quadratic equation into the form left side $=0$ , and then factorize the left side into $(x+a)(x+b)=0$ ?

$$(x+a)(x+b)=0$$

For the product of two terms to be 0, either $x+a=0$ or $x+b=0$ must be true. This means...

$$x+a=0 \quad \text{\scriptsize or} \quad x+b=0$$

So we get the solutions $x=-a$ or $x=-b$ .

Below, we've prepared a graph of the quadratic function corresponding to the quadratic equation $(x+a)(x+b)=0$ . The initial values are set to $a=1,\,b=-2$ , drawing the graph of the quadratic function $y=(x+1)(x-2)$ . Try changing the values of $a$ and $b$ to see how the shape of the graph changes. How $a$ and $b$ appear on the graph when factorized into the form $y=(x+a)(x+b)$ will also be dealt with in quadratic functions.

Using Square Roots

Using the Quadratic Formula

The Process of Solving